qr code generator vb.net Sinusoidal Voltages and Currents as Vectors in Visual Studio .NET

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Sinusoidal Voltages and Currents as Vectors
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A generator of dc voltage is usually represented by the battery symbol, Fig. 88, while an ac generator is usually represented by the slip rings symbol of Fig. 89.* In the dc case of Fig. 88, the polarity DOES NOT CHANGE WITH TIME; thus, in Fig. 88, one of the battery terminals will always be POSITIVE with respect to the other terminal. The situation can be indicated either by the use of and signs or by means of a voltage arrow placed alongside, with the understanding that the head of
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* This originated as a symbol for the circular copper slip rings used to connect the rotating coils (armature) of an ac generator to outside stationary circuitry.
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CHAPTER 5 Sinusoidal Waves. rms Value
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Fig. 88
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Fig. 89
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the arrow is POSITIVE with respect to the tail, as shown. That is, a voltage arrow will always point FROM THE NEGATIVE TERMINAL TO THE POSITIVE TERMINAL of a generator. It s obviously not necessary to show both polarity marks and voltage arrow, and thus generally, from now on, we ll use only voltage arrows, having the meaning just described above. In our work with dc circuits, in Chap. 4, we found that it is absolutely necessary to indicate the POLARITIES of the dc generators in a network. The importance of this requirement is illustrated in the simple circuits shown in Fig. 90, in which batteries represent dc generators with polarities indicated by voltage arrows as shown. (The Greek letter , capital omega, denotes resistance in ohms.)
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Fig. 90
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Notice that the two circuits give completely di erent values of current; this is because in the left-hand diagram the two dc generators are connected so as to OPPOSE each other, while in the right-hand diagram they are connected so as to AID each other. This can be understood by tracing around both circuits in the clockwise sense, remembering that going through a generator with the voltage arrow represents a rise in potential, while going through against the arrow represents a decrease in potential. The situation in Fig. 90 is represented graphically in vector diagram form below.
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The above example illustrates the fact that two dc generators in the same circuit will either totally AID each other or totally OPPOSE each other, with no in between
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CHAPTER 5 Sinusoidal Waves. rms Value
conditions possible. Thus, the voltages produced by two dc generators cannot be 458 apart, or 658 apart, or 1258 apart, and so on. Likewise, the phase angle of the voltage drops across two resistances in a dc network can have only the relative value of 08 or 1808. Now, however, consider the case of two series-connected AC GENERATORS of the same frequency. Here it s possible to have two generators in which the phase angle between their voltage waves can be ANY POSITIVE OR NEGATIVE ANGLE FROM 08 TO 3608. Likewise, the voltage drop across a circuit component can have di erent phase angles relative to the voltage drops across other circuit elements and generator voltages. Thus, unlike dc circuit analysis, in ac work we must take the factor of PHASE into account. For this reason the algebra of ac circuits is more complicated, in its inner details, than the algebra of dc circuits. Broadly speaking, however, we ll nd that many of the basic procedures we learned in dc analysis will carry over directly to ac work. With the above in mind, let us now turn our attention to the case where the voltage sources are AC GENERATORS instead of dc generators. We begin our discussion with Fig. 91, using the ac generator symbol of Fig. 89 with voltage arrows as previously mentioned. The circles labeled V1 , V2 , and V represent ac voltmeters, with I being an ac ammeter. It is given that ac meters are always calibrated to read rms values, unless de nitely stated otherwise on the face of the meter. Let us now make a careful study of Fig. 91.
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